Payne gas furnace - blower fan running continuously

Payne model 376B size 048095

symptoms:

Blower fan runs continuously no matter what the t-stat is set to temp-wise
.
Thermostat is apparently OK. I have 2 furnaces in here; the other furnace is a smaller Payne (376B 036075). Both have same Honeywell T-8200. I swap the two t-stat units and they both work when connected to the 036075 but neither works with the 048095. The t-stat connected shows no LEDs on while blower running, while the good furnace always shows amber LED when running.

The fusible links running to the limit switch: the closest 2 of the 3 are black and crispy; the 3rd one looks OK in its transparent sleeve. On the working furnace, those fusible links are all 3 OK in their transparent sleeves.

a multimeter test of the limit switch shows continuity while the furnace is unplugged. (I'm nervous of probing around with live voltages, even at just 120v)

I replaced the filters above the blower. They were absolutely filthy as if they had not been cleaned in several years!